Problem: Here's a partially-filled Hessian matrix. $\begin{bmatrix} 0 & ??? & -2zy^2 \\ \\ -2z^2y & -2z^2x & -4xyz \\ \\ -2zy^2 & -4xyz & -2xy^2 \end{bmatrix}$ What is the missing entry? Choose 1 answer: Choose 1 answer: (Choice A) A $-4xyz$ (Choice B) B $-2zy^2$ (Choice C) C $-2z^2y$ (Choice D) D There's not enough information.
Answer: The Hessian of a scalar field $f$ is the matrix that contains all its second-order partial derivative information. $\bold{H}(f) = \begin{bmatrix} f_{xx} & f_{xy} & f_{xz}\\ \\ f_{yx} & f_{yy} & f_{yz} \\ \\ f_{zx} & f_{zy} & f_{zz} \end{bmatrix}$ Because the order of mixed partial derivatives often doesn't matter, the Hessian matrix is usually symmetric. We can use this fact to find $f_{xy}$, which is equal to $f_{yx}$. Matching to the left middle of matrix, the missing entry is $-2z^2y$.